One may think of applying already known density methods on an interval \([a,b]\)  and identify this as the circle. However, one must appreciate that in order to equate the circle with an interval, it is neccessary to assign a wrapping at the endpoints of the interval. This means one can not simply employ general density methods discussed in, for example, [1]}. Boundary conditions on \(f_h\)  and \(p\)  must be obtained for these methods to be applicable — particularly, \(\lim _{x \rightarrow a+}f(x)p(x) = \lim _{x \rightarrow b-} f(x)p(x)\) . The von-Mises, Bingham and uniform distributions do not satisfy this boundary condition unless one restricts the function space for which the operator is defined on. Instead, we will modify the density approach to accommodate the geometry of \(\mathbb {S}^1\) , and we shall see that by the definition of continuous functions on \(\mathbb {S}^1\) , this condition is always satisfied for absolutely continuous \(f\)  and \(p\) . Even though one may use an interval of arbitrary length to construct the circle, we shall, in this paper, use intervals of lengths \(2\pi \)  for simplicity.
